In usual a method to discretize partial differential equation is required to have some consistency with the solution to original differential equation. When we solve hyperbolic conservation laws numerically by difference approximation, it is very important to know how consistent with the entropy solution the difference approximation or numerical result is. In the case of scalar conservation law, we know that a difference approximation converges to the entropy solution if it satisfies some condition which is described in terms of numerical viscosity coefficients. (See [1, 2, 10, 13].) In these works, the consistency with entropy condition is an important subject and is discussed through the numerical entropy inequality or cell entropy inequality, which is a discrete analogue of the entropy inequality that is employed in the definition of entropy condition by Lax. (See [6, 12].) On the other hand, the results on convergence may lack practical information on the numerical behavior of difference approximation, where the numerical behavior means the behavior of numerical result which is calculated at some finite values of difference increments (Δx, Δt etc.). But information on numerical behavior is important when we are interested in the quality of numerical computation.
We here discuss the consistency with entropy condition from the viewpoint of Oleǐnik's E-condition. Although the discussion is not applied so widely as that based on the numerical entropy inequality, the result gives more information on the numerical behavior of difference approximations. It helps us discuss the quality of numerical computation.